This paper addresses a significant class of strongly nonlinear second-order differential equations arising in heat-conduction and diffusion problems. The specific boundary value problem considered is described by: y'' = f(x, y, y'), 0 < x < 1, y(0) = 0, y(1) = 1, where f is a given function, x ∈ [0, 1], and y is a function of x. Analytically solving this problem is challenging, particularly when the function f(x, y, y') is nonlinear in y. Consequently, various numerical techniques have been developed to tackle this problem. These methods include finite difference approaches, Petrov-Galerkin methods, shooting methods, spline methods, variation iteration methods, collocation methods, asymptotic approximations, and Numeral's method. This paper delves into the analysis and application of these numerical methods for solving the given strongly nonlinear boundary value problem. The goal is to provide insights into the efficacy of these approaches and their suitability for different scenarios. Understanding the nuances of these methods is crucial for tackling a wide range of practical problems in heat-conduction and diffusion.